On a two-parameter Yule-Simon distribution Erich Baur ♠ Jean Bertoin ♥ Abstract We extend the classical one-parameter Yule-Simon law to a version depending on two pa- rameters, which in part appeared in [1] in the context of a preferential attachment algorithm with fading memory. By making the link to a general branching process with age-dependent reproduction rate, we study the tail-asymptotic behavior of the two-parameter Yule-Simon law, as it was already initiated in [1]. Finally, by superposing mutations to the branching process, we propose a model which leads to the full two-parameter range of the Yule-Simon law, generalizing thereby the work of Simon [20] on limiting word frequencies. Keywords: Yule-Simon model, Crump-Mode-Jagers branching process, population model with neutral mutations, heavy tail distribution, preferential attachment with fading memory. AMS subject classifications: 60J80; 60J85; 60G55; 05C85. 1 Introduction The standard Yule process Y = (Y (t))t≥0 is a basic population model in continuous time and with values in N := 1, 2,... It describes the evolution of the size of a population started { } from a single ancestor, where individuals are immortal and give birth to children at unit rate, independently one from the other. It is well-known that for every t 0, Y (t) has the geometric −t ≥ distribution with parameter e . As a consequence, if Tρ denotes an exponentially distributed random time with parameter ρ > 0 which is independent of the Yule process, then for every k N, there is the identity ∈ ∞ P(Y (T )= k)= ρ e−ρt(1 e−t)k−1e−tdt = ρB(k, ρ + 1), (1) ρ − Z0 arXiv:2001.01486v1 [math.PR] 6 Jan 2020 where B is the beta function. The discrete distribution in (1) has been introduced by H.A. Simon [20] in 1955 and is nowadays referred to as the Yule-Simon distribution with parameter ρ. It arises naturally in preferential attachment models and often explains the occurrence of heavy tail variables in stochastic modeling. Indeed, the basic estimate B(k, ρ + 1) Γ(ρ + 1)k−(ρ+1) as k , ∼ →∞ implies that the Yule-Simon distribution has a fat tail with exponent ρ. ♠Bern University of Applied Sciences [email protected] ♥Institut fur¨ Mathematik, Universit¨at Zurich.¨ [email protected] 1 The present work is devoted to a two-parameter generalization of the Yule-Simon distribu- tion, which results from letting the fertility (i.e. the reproduction rate) of individuals in the population model depend on their age. Specifically, imagine that now the rate at which an individual of age a 0 begets children is e−θa for some fixed θ R. So for θ > 0 the fertility ≥ ∈ decays with constant rate θ as individuals get older, whereas for θ < 0, the fertility increases with constant rate θ. Denote the size of the population at time t by Y (t). In other words, − θ Yθ = (Yθ(t))t≥0 is a general (or Crump-Mode-Jagers) branching process, such that the point process on [0, ) that describes the ages at which a typical individual begets a child is Poisson ∞ −θt with intensity measure e dt. For θ = 0, Y0 = Y is the usual Yule process. Definition 1.1. Let θ R and ρ > 0. Consider Y as above and let T be an exponential ∈ θ ρ random time with parameter ρ > 0, independent of Yθ. We call the law of the discrete random variable Xθ,ρ := Yθ(Tρ) the Yule-Simon distribution with parameters (θ, ρ). A key difference with the original Yule-Simon distribution, which corresponds to θ = 0, is that no close expression for the two-parameter distribution is known1. Actually, the general branching process Y is not even Markovian for θ = 0, and its one-dimensional distributions are θ 6 not explicit. This generalization of the Yule-Simon distribution has recently appeared in [1] for θ> 0 and ρ> 1, in connection with a preferential attachment model with fading memory in the vein of Simon’s original model. We shall point out in Section 5 that the range of parameters θ 0 and ρ> 0 arises similarly for a family of related models. ≤ One of the purposes of the present contribution is to describe some features of the two- parameter Yule-Simon law, notably by completing [1] and determining the tail-asymptotic be- havior of Xθ,ρ. It was observed in [1] that the parameter θ = 1 is critical, in the sense that when θ< 1, X has a fat tail with exponent ρ/(1 θ), whereas when θ> 1, some exponential θ,ρ − moments of positive order of Xθ,ρ are finite. We show here in Section 4 that when θ > 1, the tail of X is actually decaying exponentially fast with exponent ln θ 1 + 1/θ. Further, in the θ,ρ − critical case θ = 1, we show that X1,ρ has a stretched exponential tail with stretching exponent 1/3. By superposing independent neutral mutations at each birth with fixed probability 1 p − ∈ (0, 1) to the classical Yule process, the original Yule-Simon law with parameter ρ = 1/p captures the limit number of species of a genetic type chosen uniformly at random among all types, upon letting time tend to infinity. This fact is essentially a rephrasing of Simon’s results in [20]. We give some (historical) background in Section 5 and extend Simon’s observations to more general branching processes, for which the two-parameter distribution from Definition 1.1 is observed. In a similar vein, the number of species belonging to a genus chosen uniformly at random has been studied for generalized Yule models in several works by Lansky, Polito, Sacerdote and 1 Although the probability P(Xθ,ρ = 1) can easily be computed in terms of an incomplete Gamma function, the calculations needed to determine P(Xθ,ρ = k) for k ≥ 2 become soon intractable. 2 further co-authors, both at fixed times t and upon letting t . For instance, in [12], the linear →∞ birth process governing the growth of species is replaced by a birth-and-death process, whereas in [13], a fractional nonlinear birth process is considered instead. Both works are formulated in the framework of World Wide Web modeling. Recently, Polito [17] changed also the dynamics how different genera appear, leading to a considerably different limit behavior. The rest of this article is organized as follows. In the following Section 2, we analyze the branching process Yθ introduced above and study its large-time behavior. In Section 3, we develop an integral representation for the tail distribution of the two-parameter Yule-Simon law, which lies at the heart of our study of the tail asymptotics of Xθ,ρ in the subsequent Section 4. This part complements the work [1] and contains our main results. In the last Section 5, we relate the generalized Yule-Simon distribution to a population model with mutations, in the spirit of Simon’s original work [20]. 2 Preliminaries on the general branching process Yθ The purpose of this section is to gather some basic features about the general branching process Yθ that has been described in the introduction. We start with a construction of Yθ in terms of a certain branching random walk. Specifically, we consider a sequence Z = (Z ) of point processes on [0, ) which is n n≥0 ∞ constructed recursively as follows. First, Z0 = δ0 is the Dirac point mass at 0, and for any n 0, Zn+1 is obtained from Zn by replacing each and every atom of Zn, say located at z 0, ≥ z z ≥ by a random cloud of atoms z +ωz N , where ωz N is the family of atoms of a Poisson point { i }i=1 { i }i=1 measure on [0, ) with intensity e−θtdt and to different atoms z correspond independent such ∞ Poisson point measures. In particular, each Nz has the Poisson distribution with parameter 1/θ when θ > 0, whereas Nz = a.s. when θ 0. If we now interpret [0, ) as a set of times, the ∞ ≤ ∞ locations of atoms as birth-times of individuals, and consider the number of individuals born on the time-interval [0,t], ∞ Y (t) := Z ([0,t]) , t 0, θ n ≥ nX=0 then Yθ = (Yθ(t))t≥0 is a version of the general branching process generalizing the standard Yule process that was discussed in the introduction. We readily observe the following formula for the first moments: Proposition 2.1. One has for every t 0: ≥ (e(1−θ)t θ)/(1 θ) if θ = 1, E(Yθ(t)) = − − 6 ( 1+ t if θ = 1. −θt Proof. By definition, the intensity of the point process Z1 is e dt, and by the branching property, the intensity of Zn is the n-th convolution product of the latter. Considering Laplace 3 transforms, we see that for any q > 1 θ: − ∞ ∞ ∞ −qt −qt q E (Yθ(t)) e dt = q e E (Zn([0,t])) dt 0 0 ! Z Z nX=0 ∞ ∞ −qt = E e Zn(dt) n=0 Z0 X∞ = (θ + q)−n n=0 X θ + q = . θ + q 1 − Inverting this Laplace transform yields our claim. Remark 2.2. The calculation above shows that a two-parameter Yule-Simon variable Xθ,ρ, as in Definition 1.1, is integrable if and only if θ + ρ> 1, and in that case we have θ + ρ E(X )= . θ,ρ θ + ρ 1 − Proposition 2.1 ensures the finiteness of the branching process Yθ observed at any time. Further, it should be plain that the atoms of the branching random walk Z (at all generations) occupy different locations.